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MILLER’S 



BUSINESS ARITHMETIC 


7 


AND 


cIJf-|(nslrndoi! lig ^anccllatioiv 

WITH EXPLANATORY NOTES. 

BY 

J\ HI. 

Of Cuthbert, Randolph Co., Ga. 


CUTHBERT, GEORGIA. 

1887 . 


J. W. BURKE A CO., PRINTERS AND BINDERS, MACON, GA. 





























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MILLER’S 


BUSINESS ARITHMETIC 


AND 


Self-Instructor by Cancellation. 


WITH EXPLANATORY NOTES. 


MILLER, 

* V 

Of Cuthbert, Randolph County, Ga. 





CUTHBERT, GA. 
1887 . 





Entered according to Act of Congress, in the year 1887, by 
J. H. MILLER, 

In the Office of the Librarian of Congress, at Washington. 
BIGHT OF TRANSLATION RESERVED. 


Q. 0- Vi i 

Westcott & Thomson, 
Stereotypers and Electrotypers, Philada. 







PREFACE. 


Having had eighteen years’ experience in 
traveling, teaching and giving public lectures 
on the science of numbers, I have obtained 
many advantages, and in this little work pro¬ 
pose giving my fellow-men the benefit of these 
advantages by presenting a series of rules based 
on “common-sense” principles and not filled 
with mathematical brushwood. But to under¬ 
stand and appreciate its merits the student must 
have a knowledge of the fundamental rules of 
arithmetic. Brevity in the solution of problems 
is very essential and necessary and cannot be 
dispensed with in every-day business, especially 
in the present fast age. 

It will be observed that we use a vertical line 
“right and left” and abbreviate by “cancel¬ 
lation,” which as taught by the arithmeticians 
of the day is complete, though as presented is 
somewhat complicated and but the fewest num- 



4 


PREFACE. 


ber adopt it. In the present work we propose 
to use brevity and simplicity, making the same 
interesting and pleasant to the pupil. It will 
be seen that eaclf rule tells what numbers to 
place on the right and what on the left of this 
vertical line, and under the rule is a plain 
“ explanatory note ” telling the whys, etc.; 
otherwise it would amount to a mechanical 
solution of the proposition. Hence fail to give 
vent to your feelings. Indeed, any rule not 
understood is mechanical. The whole principle, 
as herein taught, is by proportion, and where 
there is no proportion there is no mathematics. 
I dwell, to some extent, on the principle of 
cause and effect, which is fully explained in my 
Common-Sense Arithmetic, brought out in 1882. 

I deem it unnecessary further to preface this 
little work, and send it out upon its own merits, 
with the sweet and consoling thought that figures 
“ properly put together ” cannot deceive the 

Author. 


Cuthbert, Ga., Mar. 1, 1887. 


CONTENTS 


PAGE 

Signs, Arithmetical. 9 

Table of Units. 10 

Table of Paper. 10 

Table of Apothecaries’ Weight. 11 

Table of Avoirdupois Weight. 11 

Table of Surveyor’s Measure. 11 

Table of Time. H 

Table of Federal Money. 12 

Table of English Money. 12 

Table of Troy Weight. 12 

Table of Square Measure...... 12 

Table of Liquid Measure. 13 

Table of Dry Measure. 13 

Table of Long Measure. 14 

Table of Cubic Measure. 14 

Table of Cloth Measure. 14 

1 c 

Fractions. 

To Reduce an Improper Fraction to a Whole or a 

Mixed Number.*. 1^ 

To Reduce a Mixed Number to an Improper Frac- 

17 

tion.. 

Remark on Mixed Numbers. I 7 

Addition of Fractions. 1® 

Subtraction of Fractions. 19 


5 

























6 


CONTENTS. 


PAGE 

Multiplication of Fractions. 20 

To Reduce a Fraction to Lowest Terms. 22 

To Multiply a Whole Number by a Fraction. 22 

Division of Fractions. 23 

To Divide a Whole Number by a Fraction.... 25 

Complex Fractions.. . 26 

To Reduce Complex Fractions to Simple Ones. 26 

Lumber Measure. 27 

For Tapering Timber. 29 

Land Measure in Rods. 30 

Land Measure in Yards. 30 

Land Measure in Chains... 31 

To Find the Area of a Triangle. 32 

To Find the Number of Cords in a Pile of Wood. 33 

To Gauge a Box or a Granary....,. 34 

To Gauge Corn in the Ear. 35 

To Find the Number of Bricks necessary to Build a 

Wall. 36 

To Find the Number of Bricks required to Pave a 

Yard or a Walk. 38 

To Find the Number of Shingles or Boards that will 

Cover a Roof. 39 

For Cutting Rafters. 40 

To Find the Number of Yards of Carpeting needed to 

Carpet a Floor. 41 

For Computing Interest. 43 

To Ascertain the Time that Elapses from One Date to 

Another. 45 

Interest from One Date to Another. . . 47 

Bank Discount. 48 

Commercial Discount. 50 




























CONTENTS. 


7 


PAGE 

To Find the Percentage on any Sum or Number at 

any Rate per cent. 51 

Having the Principal, the Time and the Amount 

Gained, to Find the Rate per cent. 53 

Having the Principal, the Rate per cent, and the 

Amount Gained, to Find the Time. 54 

Having the Time, the Rate per cent, and the Amount 

Gained, to Find the Principal. . 55 

Profit and Loss. 56 

Having the Cash- and Credit-Price of an Article, to 

Find the Gain per cent. 57 

Having the Cost of an Article, to Find how it should 

be Sold to Gain any given per cent. 50 

To Arrive at the Average Cost of a Nest of Tubs, 

Trays, Trunks, etc. 60 

Cause and Effect. 61 

Partnership. 67 

Barter. 70 

Averaging the Price of Cotton, etc. 71 

To Find the Capacity of a Tank in Gallons. 73 

To Find the Capacity of a Round Tank in Gallons 74 

Having the Length, to Find how much Land to cut 

off to give any Desired Number of Acres. 74 

Having the Length in Yards, to Find the Width re¬ 
quired for One Acre of Land in Yards. 75 

An Easy and Quick Process for Multiplying any 

Number by the Aliquot Part of 100. 76 

Mensuration. 78 

Having the Diameter of a Circle, to Find its Area. 78 

Having the Circumference of a Circle, to Find its 
Area. 70 






















8 


CONTENTS. 


PAGE 


To Find the Area of a Globe or a Ball. 80 

To Find the Solidity of a Globe or a Ball.*.. 81 

To Find the Area of the largest Square that can be 

Inscribed within a Circle... 82 

To Find the Solidity of a Cylinder. 83 

Having the Diameter of two Circles, one formed within 

the other, to Find the Area of the Ring. 84 

To Find where a Pole should be Broken to Strike the 
Ground at any Given Distance and Hang on the 

Stump. 85 

To Measure the Distance to any Given Object. 87 

Table giving Number of Plants to Acre, etc.. 89 










MILLER’S BUSINESS ARITHMETIC 


AND 

SELF-INSTRUCTOR. 


SIGNS. 

The Sign of Equality. —Two short hori¬ 
zontal lines (=), read “ equal ” or “ equal to.” 
Thus, 12 inches = 1 foot. 

The Sign of Addition. —An erect cross 
(+), read “plus” or “added to.” It denotes 
that the quantities between which it is placed 
are to be added together. Thus, 4 + 8 signifies 
that 4 is to be added to 8, =12. Placed after 
an answer, it also shows there is a remainder. 

The Sign of Subtraction.—A short hori¬ 
zontal line (—), read “minus” or “less.” It 
denotes that the quantity on the right is to be 
subtracted from the quantity on the left. Thus, 
12—4 signifies that 4 is to be subtracted from 
12 , = 8 . 

The Sign of Multiplication. —An in- 

9 



10 miller’s business arithmetic. 

dined cross (x), read “ times” or “multiplied 
by.” It denotes that the quantities between 
which it is placed are to be multiplied. Thus, 
4x8 signifies that 4 is to be multiplied by 8, 
= 32. 

The Sign of Division.—A horizontal line 
placed between two dots (-*-), read “ divided 
by.” It denotes that the quantity on the left 
is to be divided by that on the right. Thus, 
25-^5 signifies that 25 is to be divided by 
5, = 5. 


Table of Units. 


12 units make 1 dozen (marked 
12 dozen “ 1 gross ( “ 

144 “ “ 1 great gross ( “ 

20 units “ 1 score ( “ 

200 lbs. “ 1 barrel of pork. 

196 “ “ 1 “ flour. 


doz.). 

gro.). 

gg-)- 

SC.) 


Paper. 

24 sheets make 1 quire. 

20 quires “ 1 ream. 

2 reams “ 1 bundle. 

5 bundles “ 1 bale. 


TABLES. 


11 


Apothecaries’ Weight. 

20 grains make 1 scruple (marked sc.). 

3 scruples “ 1 dram ( “ dr.). 

8 drams “ 1 ounce ( “ oz.). 

12 ounces “ 1 pound ( u lb.). 

Note. —This table is used by druggists in compounding 
medicines. 

Avoirdupois Weight. 

16 drams make 1 ounce (marked oz.). 

16 ounces “ 1 pound ( “ lb.). 

2000 pounds “ 1 ton ( “ T.). 

Note. —This table is used in weighing iron and cotton 
and in loading vessels. 


Surveyor’s Measure. 

7^^- inches make 1 link (marked 1.). 

100 links “ 1 chain ( “ ch.). 

80 chains “ 1 mile ( u m.). 


Table of Time. 

60 seconds make 1 minute. 

‘ 1 hour. 

‘ 1 day and night. 

‘ 1 month. 


60 minutes 
24 hours 
30 days 
12 months 


year. 


12 calendar months 1 1 Julian or common 
or 365 days / year. 

Note. —360 days make a commercial year. 


12 miller’s business arithmetic. 



Federal Money. 


10 mills 

make 1 cent (marked ct.). 

10 cents 

« 

1 dime ( “ 

dm.). 

10 dimes 

« 

1 dollar ( “ 

•). 

12J cents 

u 

J of 1 dollar. 


16f “ 

(( 

1 U 1 (( 

6 1 


25 “ 

u 

i “ 1 “ 


33J “ 

« 

i “ 1 “ 


37* “ 

(( 

f “ 1 “ 


50 “ 

u 

i “ 1 “ 


62J “ 

a 

I “ 1 “ 


75 “ 

u 

f “ 1 “ 


87i « 

a 

l “ 1 “ 



English Money. 


4 farthings 

make 1 penny (marked d.). 

12 pence 

u 

1 shilling ( “ 

s.). 

20 shillings 

(( 

1 pound ( “ 

£)- 

21 “ 

(( 

1 guinea ( “ 

g-)- 


1 pound sterling is equal to $4.84. 

1 shilling “ .24J- cts. 

1 penny “ .02-^g- cts. 

Troy Weight. 

24 grains make 1 pennyweight (dwt.), 

20 pennyweights “ 1 ounce (oz.). 

12 ounces “ 1 pound (lb.). 

Note. — All jewelry is weighed by this weight. 


TABLES. 


13 


Square Measure. 

144 square inches make 1 square foot (sq. ft.). 


9 square feet “ 
30J square yards “ 

40 rods “ 

4 roods “ 

4840 square yards “ 

160 square rods “ 

10 square chains “ 

640 square acres “ 

Note. —The above is used 


1 square yard (sq.yd.). 
1 square rod or pole 
(sq. rd.). 

1 rood (r.). 

1 acre (a.). 

1 acre. 

1 acre. 

1 acre. 

1 mile (sq. m.). 
i surface measure. 


Liquid Measure. 


4 gills 

make 1 pint (marked pt.). 

2 pints 

“ 1 quart ( “ 

qt.). 

4 quarts 

“ 1 gallon ( “ 

gal.). 

63 gallons 

“ 1 hogshead ( “ 

hhd.). 

231 cubic inches make 1 gallon. 



Dry Measure. 


2 pints 

make 1 quart (marked qt.). 

8 quarts 

“ 1 peck ( “ 

pk.). 

4 pecks 

“ 1 bushel ( “ 

bu.). 

5 bushels 

u 1 barrel ( “ 

bbl.). 


2150^^ cubic inches is equal to 1 bu v and 
is the U. S. standard. 


1 1 miller’s business arithmetic. 
Long Measure. 


12 inches make 1 foot 

(marked ft.). 

3 feet 1 

u 1 yard 

( 

« 

y d -)- 

5J yards 

“ 1 rod 

( 

u 

rd.). 

40 rods 

“ ' 1 furlong ( 

u 

fur.). 

8 furlongs 

“ 1 mile 

( 

u 

m.). 

3 miles 

“ 1 league 

( 

<( 

lg-)- 

69^- miles 

“ 1 degree 

( 

u 

de g-)- 


360 degrees make the circumference of the 
earth. 

4 inches make 1 hand, used in measuring the 
height of horses. 

6 feet make a fathom, used to measure the 
depth of water. 

Cubic Measure. 

1728 cubic inches make 1 cubic foot. 

27 cubic feet “ 1 cubic yard. 

40 feet of round or 50 feet1 ^ 
of hewn timber / 

42 solid feet make 1 ton of shipping. 

128 cubic feet make 1 cord of wood. 

Cloth Measure. 

2J inches make 1 nail (marked na.). 

4 nails “ 1 quarter ( “ qr.). 

4 quarters “ 1 yard ( “ yd.). 


FRACTIONS. 


15 


FRACTIONS. 

Presuming the pupil to be familiar with the 
fundamental rules of arithmetic, we commence 
our work with fractions, and do so by pro¬ 
pounding such questions as we think necessary 
in the premises: 

Question. What are fractions ? 

Answer. Broken units or numbers. 

Q. Of how many parts is a fraction com¬ 
posed ? 

A. Two; the numerator and the denomi¬ 
nator. 

Q. What does the denominator show ? 

A. The number of parts into which tL. 
is broken. 

Q. What does the numerator show ? 

A. The number of parts taken? 

Q. What is a proper fraction ? 

A. One of which the numerator is less than 
the denominator. 

Q. What is an improper fraction ? 

A. One whose numerator is greater than the 
denominator. 

Q. What should be done with improper frac¬ 
tions ? 


16 miller’s business arithmetic. 


A. They should be reduced to a whole or a 
mixed number by Rule 1. 

Rule 1. Divide the numerator by the denomi¬ 
nator. 

Prop. 1. Change to a whole or a mixed 
number. 

OPERATION- 

75-*-4= 18f. Ans . 18f. ; 

2. Change to a mixed number. 

Ans. 33-g-. 

3. Change -^ 3 - to a mixed number. 

Ans. 16^|. 

4. Change - 5 ^ to a mixed number. 

Ans. 16f. 

5. Change ^ to a mixed number. 

Ans. 5^. 

Question. What is a mixed number? 

Answer. A whole number joined or added 
to a fraction; as, 5^-, 16§, 33etc. 

Q. What should be done with mixed num¬ 
bers ? 

A. They should be reduced to improper 
fractions. 

Q. By what rule should they be so reduced ? 



FRACTIONS. 


17 


A. Rule 2. Multiply the whole number by 
the denominator of the fraction , and add in the 
numerator. 

Prop. 1. Reduce 33J- to an improper frac¬ 
tion. 


OPERATION. 

33 ^ 

100 Ans. ifa. 

3 

Note—W e see that it takes f to make the unit; in 33- 
units there are and 33£=-Mp, the answer. 

2. Reduce 16f to an improper fraction. 

Ans. 

3. Reduce 16-j-J to an improper fraction. 

Ans. 2 ^. 

4. Bring 18f to an improper fraction. 

Ans. 

5. Bring 2150^^- to an improper fraction. 

Ans. 

Remark. —All mixed numbers, wherever or whenever 
they occur, must be reduced to improper fractions by Rule 
2; after which the numerator is to be placed on the same 
side of the vertical line as the mixed number, with the 
denominator on the opposite side. 

“Notes” and “Remarks” must be observed in order 
to proceed correctly and with understanding. 

2 



18 miller’s business arithmetic. 


ADDITION OF FRACTIONS. 

Rule. Multiply all the denominators together 
for a common denominator, and all the numer¬ 
ators by each denominator except its own; then j 
add the products and place the sum over the com¬ 
mon denominator. If the numerator be greater I 
than the denominator , divide the numerator by 
the denominator. 

Prop. 1. Add ^ and f. Ans. -J = 1^. 

OPERATION. 

i x | = 6, common denominator, 

1x3 = 3 

2x2=_£ 

7 = 6 = H- Ans. 

2. T 5 2+| + -| = Iff- Ans - 1 tf- 

3. i+J- + i = H = l*- Ans.lj\. 

4. What is the sum of i+i + i + i? 

Ans. 

5. What is the sum of f + i + f ? 

Ans. 1 Y^g • 

6. What is the sum of f + f + ? 

Ans. lfj. 

Note.— When the denominators are alike, the oper 
tion may be performed by simply adding the numerato 
• and placing the result over the denominator. 





SUBTRACTION OF FRACTIONS. 


19 


Prop. 1. | +1 = If.. Ans. 1|. 

Note.—I n adding mixed numbers it is best to add the 
whole numbers and the fractions separately. 


SUBTRACTION OF FRACTIONS. 

Pule. Reduce the fractions to a common de¬ 
nominator, placing the difference of the numer¬ 
ators over the common denominator , and reduce 
the fraction to its loivest terms by Rule 1. 

Prop. 1. From f take -§. Ans. -§. 

OPERATION. 

3x8 = 24-12 = 12 
4x3=JI2 
12 

2. f — \ — xo • Ans. -fa. 

3. From \ of f of f = ^ take J of -J- of 

I = 3V* 3V ~ i = . Ans. -J-f. 

4. From 18f take 16-J. Ans. 2 fa. 

5. I bought of A. J. Baldwin & Co. 9 lbs. 

9 oz. of tobacco, and sent back 7 lbs. 3 oz.; how 
much did I retain ? Ans. 2-| lbs. 

6. On settlement I owed A. P. Stewart $4^-, 
and paid him $2|; how much remained due ? 

Ans. $2^. 



20 miller’s business arithmetic. 


MULTIPLICATION OF FRACTIONS. 

Rule 3. Place all the numerators on the right 
of a vertical line } and all the denominators on the 
Left. 

Prop. 1. Multiply § by f. 

OPERATION. 

\ ' 3 1 Ans. I 

2 I 

Note.—I n the above operation we see that the 3 on the 
left will cancel the 3 on the right one time; this causes 
the 1 to be a new numerator. * The 2 on the right will 
cancel the 4 on the left two times; the 2 then becomes a 
new denominator. Hence the answer, \. 

2. Multiply \ by \. Ans. 

3. Multiply ^ of -J of by -§-. Ans. jr. 

4. What would -J of f of yds. of goods cost, 

at -J of f dollars per yd. ? Ans. 

5. Multiply together | of f of f of f of 

Ans. -jt. 

6. What is the cost of 8f lbs. beef, at 5^ 
cts. per lb. ? 


OPERATION. 

4 j 33 9x5= Ans. 45 cts. 

NOTE. —8f = \ 5 -; 5} = - 3 ^. 



MULTIPLICATION OF FRACTIONS. 21 


7. What will 7f yds. of ribbon cost, at 12-J- 

cts. per yd.? Ans. 95 cts. 

8. Give the cost of 16f lbs. of bacon, at 8J- 

cts. per lb. ? Ans. $1.40. 

9. What cost 17^- sks. of brand, @ $3f per 

sk. ? Atis. $63. 

10. What cost 18f yds. muslin, @ 18f cts. 

per yd.? Ans. $3.51 T 9 g-. 

11. Give the cost of -J of f of 50 lbs. coffee, 

at 12f cts. per lb. Ans. $3.15. 

12. What would be the cost of 4^- of 4-| of 
6J of 4-| lbs. of beef, at 2J cts. per lb. ? 

Ans. $14.10|. 

OPERATION. 


41 4f 6f 4f 21 

I ¥ V ¥ I 


% 

9 

4 


9 

W 

27 


9 a 
i 9 


11 X 19 X 27 = 5643 -s- 4 = 14.10| 


Note. —All the numbers on the left are canceled except 
4; hence, 4 is the divisor. In all propositions cancel all 
the numbers that can be canceled without leaving a re¬ 
mainder, after which multiply together all uncanceled 
numbers on the right, all on the left together, and divide 
the product of the right by the product of the left. If 
the product of the left be greater than the product of the 
right, it will only be “ a proper fraction” to be reduced to 
its lowest terms. No fraction is in its lowest terms as 
long as both numerator and denominator can be divided 
by any number greater than 1 and leave a remainder. 






22 miller's.BUSINESS ARITHMETIC. 

To Reduce a Fraction to its Lowest Terms. 

Rule 4. Divide the numerator and the de¬ 
nominator by any number greater than 1 that 
will go in both without having a remainder . 

Prop. 1. Reduce T 4 ¥ to its lowest terms. 

OPERATION. 

4 I Ans ‘ i* 

2. Reduce to its lowest terms. Ans. -J-. 

3. Reduce to its lowest terms. Ans. 

4. Reduce to its lowest terms. Ans. -j-^. 

5. Reduce -| to its lowest terms. Ans. -|. 

6. Reduce to its lowest terms. Ans. 

7. Reduce A to its lowest terms. Ans. f. 

8. Reduce A to its lowest terms. Ans. J-. 

To Multiply a Whole Number by a Fraction. 

Rule 5. Place the whole number and the 
numerator of the fraction on the right, and the 
denominator of the fraction on the left. 

Prop. 1. What will 480 peanuts cost, at 
cts. each? 


OPERATION. 

m 


7x60 = 420. 


Ans. $4.20. 


DIVISION OF FRACTIONS. 


22 


2. What will 448 lbs. of cotton be worth, at 

8J cts per lb. ? Ans. $36.96. 

OPERATION. 

s ?—-\ 3 

\ 33x112 = 36.96 

3. Give the cost of 880 lbs. of meat, @ 6^ 

cts. per lb. Ans. $55.00. 

4. Multiply 10 by 3-|-. Ans. 35. 


5. Multiply 10^- by 3^-. 

6. Multiply .5-|- by 5J-. 


Ans. 36f. 
Ana. 301. 


7. What is the cost of 5^ lbs. candy, at 5|- 

cts. per lb. ? Ans. 30^ cts. 

8. What would f- of -J of -f- of of f yds. of 
goods cost, at ■£• of -J of J of f cts. per yd. ? 

Ans. ^ cts. 


DIVISION OF FRACTIONS. 

Rule 6. Place the numerator's of the dividend 
on the right and the numerators of the divisors 
on the left , with the denominators on the opposite 
side. Or, place on the right all the numerators 
preceding the word “ by,” those following on the 
left , with the denominators on the opposite side . 

Note.— The rule within itself: Invert the terms. 

Prop. 1. Divide £ by \. Ans. 2. 

OPERATION. 

%\l 



24 miller’s business arithmetic. 


2. Divide f of £ by £ of £. Arts. |. 

3. Divide j by f. Arts. If 

4. Divide & of |f of f of f by ff of ff 

Ans. 

5. Divide | of £ of f of £ of £ by f of £. 

6. Divide f of | of of ^ by -fy of f£. 

u4?is. 

7. Divide f X f X f by £ X -X f. 

1-g-, 

8. Divide 4f = 4° by f of T 4p .4ns. 30. 

9. I bought 34-J bu. of wheat, for which 1 
paid $68J; what was the cost per bu. ? 

Ans. $2. 

OPERATION. 

68} = ^;34H*p. fim 

ml p =2 

10. We bought 13£ lbs. candy for 94| cts.; 

what was the cost per lb. ? A ns. 7 cts. 

Note.—D ivide 94£ by 13^ =cost. 7 cts. 

11. Divide 18f X 6J. .4ns. 3. 

12. Divide f of | by % of f; multiply the 

quotient by f of of f ; divide the product by 
t °f f °f = ^) multiply this quotient by 
ft of 480. Ans. 64. 


DIVISION OF FRACTIONS. 


25 


13. What number multiplied by ^ of of |- 

will produce 1000? Ans. 4000. 

Note.— Divide 1000 by ^ of § of f to find the number 
sought. 

14. What number multiplied by f of | will 

give 60? Ans. 240. 

To Divide a Whole Number by a Fraction. 

Rule 7.— Place the whole number and the 
denominator of the fraction on the right, and 
the numerator of the fraction on the left. 

Prop. 1. Divide 12 by f. v Ans. 16. 

operation. Note. —12 units = ^, so you divide \ 2 - 

^ [ 4'x4 = 16 

2. Divide 16 by f. Ans. 26§-. 

3. Divide 48 by 4£ = 2A. Ans. 10. 

4. Divide 100 by J-J. Ans. 200. 

5. What number multiplied by f of f of f 

will give 100? Ans. 250. 

Proof. —Multiply 250 by | of f of f and see the proof, 

OPERATION. 

3 50x2 = 100. 

i 2 

3 3 

i 



26 miller’s business arithmetic. 


COMPLEX FRACTIONS. 

These are such as have a fractional nu¬ 
merator or a fractional denominator, or some¬ 
times both. Fractions denote division. The 
numbers above the line are to be regarded as 
the dividends, while those below are the di¬ 
visors, and the example is to be worked out as 
in the division of fractions, under Rule 6. 

For instance, what part of a foot is 6f in. ? 

6f 

It equals ; then we regard the 6f inches as 


the dividend and the 12 as the divisor, and, 
divided, 6f — ^ by 12. 


OPERATION. 

41 27 = 3)11 = x 9 e reduced. 
121 


48; 


To Reduce Complex Fractions to Simple Ones. 

Rule 8. Consider the numerator as the divi¬ 
dend, the denominator as the divisor , and proceed 
as directed by Ride 6. 

6 ^ 

Prop. 1. Reduce 6J = - 2 ^-; then divide 

■¥- b y I- 

OPERATION. 

^ | ^ = 10 = Ans. 10 . 







LUMBER MEASURE. 


27 


8 — 

2. Reduce — to the simple fraction of a foot. 


Ans. |4. 

3| 8 

3. Reduce — to a simple fraction. Ans. -§-. 

16} 

4. Reduce — to a simple fraction. Ans. f. 

5. What part of a yard is 18} inches? 

Ans. 


Lumber Measure. 

Rule 9. Place the number of pieces, with the 
length, width and thickness, on the right; on the 
left, 12, or its factors, 3 and If. 

T*rop. 1 . How many feet in 100 planks 12 
\>ng, 9. in. wfde and 1 in. thick? 

Ans. 900 ft. 

OPERATION. 

I 100 

n\ n 

| 9x100 = 900. 

Note. —We use the 12 (or its factors) on the left to 
reduce it to feet. Were we to multiply the length in inches 
it would give 144 times too much. To bring the contents 
in feet for board measure, we divide by 12. 

2. How many feet in 400 pieces 18 ft. long 
6 in. wide and 4 in. thick ? Ans. 14400. 



28 miller’s business arithmetic. 

3. What number of feet in 144 planks 16 ft. 
long, 6 in. wide, 1 in. thick? Ans. 1152 ft. 

4. Give the number of feet in 36 planks 12 
ft. long, 8 in. wide, 1 in. thick. Ans. 288 ft. 

5. What number of feet in 20 pieces of lum¬ 
ber 36 ft. long, 5 in. wide, 3 in. thick ? 

Ans. 900 ft. 

6. Give the number of feet in 25 scantling 15 

ft. long, 4 by 3 in. Ans. 375 ft. 

7. How many feet in 12 planks 12^- ft. long, 
12J in. wide and 2 in. thick? Ans. 312^- ft. 

8. How many feet in 120 pieces of lumber 
11^- ft. long, 9J in. wide and 1 T 9 T in. thick? 

Ans. 2000 ft. 

9. How many feet in, and what would be 
the cost of, 32 planks 16f ft. long, 6f in. wide 
and 1 in. thick, at fl-J per hundred feet? 

Ans. 300 ft.; cost, $5. 

10. How much lumber in, and what is the 
cost of, 20 pieces 7 ft. long and 4 by 6 in., at 
$2^- per hundred feet? Ans. 280 ft.; cost, $7. 

11. How many feet in 800 pickets 12 ft. 
long, 1-1- in. wide and \\ in. thick? 

Ans. 1500 ft. 


FOB TAPERING TIMBER. 


29 


For Tapering Timber. 

Rule 10 . Add together the areas of the two 
ends , and place the sum , also the length , on the 
right; on the left , 2, 12. 

Note. —The 2 denotes a mean of the areas of the two 
ends, and the 12 is to reduce to feet for board measure. 

The same rule will produce a correct result whether 
there be any taper or not. 

Prop. 1 . How many feet board measure in 
a stick of timber 40 ft. long, 12 X 12 in. at the 
large end and 4 X 4 in. at the small ? 


OPERATION. 

12 x 12 = 144, area large end, 

4x 4= 16, area small end, 

160 = united areas. 

% I m 

3 0 

4 10x80= 800-3 = Ans. 266 

4X4 

2. How many feet board measure in 20 scant¬ 
ling 20 ft. long, 4X5 in. at the large end and 
2 X 3 in. at the small end ? Ans. 433^ ft. 

3. Give the contents in board measure of a 
piece of timber 10 X 10 in. at one end and 
6X6 in. at the other, and 20 ft. long. 

Ans. 133 ft. 







30 MTLLER’s BUSINESS ARITHMETIC. 


4. How many feet in a ship’s mast, 20 X 20 
in. at the large end and 4X4 in. at the other, 
60 ft. long? and what would be the cost at $1-1- 
per hundred ft. ? Arts. 1040 ft.; cost, $15.60. 


Land Measure in Rods. 

Rule 11 . Place the length and the width, in 
rods, on the right; on the left, 160, or its factors, 

4, io, 4 . 

Prop. 1. How many acres in a field 80 rds. 
long and 40 rds. wide ? Ans. 20 acres. 

operation. Note. —The length and the width in rods 

I ffl on l he right equal the number of square rods 
% | W = 20 j n the geld \y e use 160 or its factors as a 
divisor to obtain acres: 160 sq. rds. = 1 acre. 


2. How many acres in a lot of land 180 rd. 

•each way ? Ans. 2021 A. 

3. What is the area of a field in acres, 160 
rd. in length and 40 rd. wide ? Ans. 40 A. 

4. We have a patch 40 rd. in length and 4 
rd. in width; how many acres? Ans. 1 A. 

Land Measure in Yards. 

Rule 12 . Place the length and the width, in 
yatds, on the right; on the left, 1^8If), or its fac¬ 
tors, 11, 11, 10, 





LAND MEASURE IN CHAINS. 


31 


Prop. 1. How many acres in a tract of land 
1100 yds. each way ? Ans. 250 A. 


XX 

xx 

n 

x 


OPERATION. 

xxw m io 
im m 

25x10 = 250. 


Note. —11, 11, 10, 4 = 4840; then, as 4840 sq. yd. = l 
A., we use it on the left as a divisor. 


2. How many acres of land are embraced in 
the right of way of the S. W. R. R. from Macon 
to Eufaula, 140 m., and the right of way 50 
yds. ? Ans. 2545^- A. 

Note. —Reduce the length to yards and work under 
Rule 12. 

operation. 


11 

XX 

XV 

X 


•140 miles 
XXfiV yds. = 1 mile 
50 “ = right of way 

m 

XX 4 X 50 X 140 = 28000 = 11 = 2545 X 5 T . 


Land Measure in Chains. 

Rule 13. Place the length and the width, in 
chains, on the right, and 10 on the left 

Prop. 1. How many acres in a plat of land 
40 ch. long and the same in width ? 

Ans. 160 A. 




32 


miller’s business arithmetic. 




OPERATION. 


40x4 = 160. 

Note.— 10 sq. ch. is equal to 1 A.; hence, we use it as 
the divisor. 


To Find the Area of a Triangle. 

Rule 14. Multiply half the altitude by the 
base, or half the base by the altitude, and place 
the product on the right; on the left, the standard 
of acres. If yards, 11, 11, 10, 4 > tf rods, 4, 
10, 4; if chains, 10. 



Prop. 1. How many acres 
in a triangle, base 80 rd., 
altitude 120 rd.? 


OPERATION. 


£ the alt. 
Base 


60 n 

so ;p 

4800 X 


xm 

W 30 


A ns. 30 A. 


2. We have a small field of a triangular shape; 
the base is 36 rd.; altitude, 90 rd. What is its 
area in acres? Ans. 10-§- A. 


Note.—T he area of any plat of land of a triangular 
shape—narrow at one end and broad at the other—can be 
obtained by adding the two ends together and dividing by 
2; this will give the average width. 








NUMBER CORDS IN PILE OF WOOD. 33 


Note. —If the field is 160 rods long 
and 80 rods in width at one end and 
20 rods at the other, we see the length 
= 160 rd.; and when we add the two 
ends, 80 + 20 = 100, half of which is 
50; so the mean width is 50 rods. 

Ans. 50 A. 

OPERATION. 

i m w 

W 50 = Ans. 


20 rods. 



3. John M. Fulton has a field 46|- rd. wide 
at the north end, 81f rd. at the south end, and 
90 rd. in length. What is its area in acres ? 

Ans. 36 A. 


Note. —We cannot impart a knowledge of surveying 
in a work like the present, though the rules given for 
land measure, together with “common sense,” will enable 
us to get the area of a plot of land of almost any ordinary 
shape. Hence we deem it unnecessary to dwell longer 
upon the subject of Land Measure. Rule 39 will be found 
convenient for the farmer. 


To Find the Number of Cords in a Pile of Wood. 

Rule 15. Place the length , the width and the 
height of the pile , in feet, on the right; on the 
left, J, 8, J. 

3 






34 miller’s business arithmetic. 


Prop. 1. How many cords in a pile of wood 
16 ft. long, 8 ft. broad and 4 ft. high ? 

Ans. 4 cds. 


OPERATION. 


4 

? 

4 


10 = 4 
* 

4 


Note.— The 4, 8, 4 on the left are the factors of 128, 
there being 128 cu. ft. in a cord. In other words, a pile 
of wood 8 ft. long, 4 ft. broad and 4 ft. high contains 1 
cord of wood. 


2. H. B. Elder has a house 40 ft. long, 16 
ft. wide and 8 ft. high; how many cords of 
tanbark can be put in said house? 

Ans. 40 cds. 

3. How many cords of bark can Harper 
Black store away under his shed, which is 6 Z 
ft. long, 24 ft. wide tpid 12 ft. high? and what 
would be the cost of the same, at $lf per cd. ? 

Ans. 144 cd.; cost, $252. 

4. Give the number of cords in a pile of 
wood 33^ ft. long, 8^- ft. broad and \\ ft. high. 

Ans. 9|ff cds. 

To Gauge a Box or a Granary. 

Rule 16. Place the length , the width and the 
depth of the box , in feet , and on the right; on 
the left , 5. 





TO GAUGE CORN IN THE EAR. 


35 


Prop. 1 . How many bushels of wheat will 
be held by a box which is 5 ft. long, 4 ft. broad 
and 4 ft. deep ? Ans. 64 bu. 

Note. —We allow cu. ft* to the bushel, which is not 
precisely correct, though very near it. We divide the 
number of cubic feet in the box by 14 , which = f; so this 
gives the 5 on the left and the extra 4 on the right. 

2. How many bushels of wheat can Capt. 
J. J. Harper put in a wheat-house which is 10 
ft. long, 4 ft. broad and 3 ft. deep? Ans. 96 bu. 


To Gauge Corn in the Ear. 

Pule 17. Place the length , the width and the 
depth of the barn , in feet , with 2, on the right , 
and 5 on the left. 

Prop. 1 . How many bushels will be con¬ 
tained by a barn which is 20 ft. long, 15 ft. 
wide and 8 ft. deep ? Ans. 960 bu. 


OPERATION. 


20 


J* 


8 

2x3 


x 8 x 20 = 960 bu. 


Note. —For corn in the ear, with no shuck, we allow 
2| cu. ft. to the bushel, though if in the shuck such de¬ 
duction can be made as is thought necessary after seeing 
the corn. The rule is offered only as a general rule, as it 



36 miller’s business arithmetic. 

would be impossible to furnish a correct one for corn in 
bulk. The rule is based upon the principle that 2 bushels 
of corn in the ear, with no shuck, is equal to 1 bushel 
shelled. 

2. My neighbor Mr. John Mattox has a 
barn 20 ft. long, 18 ft. wide and 10 ft. deep; 
how many bushels of unshucked corn will it 
hold, allowing £ off for shuck? 

Ans. 960 bu. 

Note.— If it be desired to obtain the result in barrels, 
put an extra 5 on the left, as 5 bu. = 1 bbl. 

3. How many bushels in a crib 15f ft. long, 
7-J- ft. high and 6J ft. deep, provided it be full 
of corn with no shuck ? and how many bbls. ? 

Ans. 302f bu.; 60^ bbls. 

To Find the Number of Bricks necessary to 
Build a Wall. 

Rule 18. Place the length, the height and the 
thickness of the wall, in feet, with 1728, on the 
right; on the left place the length, the width and 
the thickness of the bricks, in inches. 

Prop. 1. How many bricks 8 in. long, 4 in. 
broad and 2 in. thick will be required to build 
a wall 800 ft. long, 4 ft. high and 2 ft. thick ? 

Ans. 172800 bricks. 


BRICKS TO BUILD A WALL. 


37 


OPERATION. 


* 


m 

i 

% 

1728x100 = 172800. 


Note. —The number 1728, on the right, is used to bring 
the cubic contents of the wall to cubic inches, which must 
be done, as the dimensions of the bricks are in inches on 
the left. 


2. What number of bricks 8 in. long, 4 in. 
wide and 2 in. thick will be required to build a 
wall 240 ft. long, 6 ft. high and 2 ft. thick ? 

Aixs. 77760 bricks. 

3. Required the number of bricks 8^ in. 

long, 4J in. wide and 2 ^ in. thick necessary to 
build a wall 933J ft. long, 6^- ft. high and 2^ ft. 
thick. Aixs. 269568 bricks. 

4. How many bricks 8*X4X2 in. will be 
required to build a house 80 ft. long, 40 ft. 
wide and 20 ft. high, the walls to be 2 ft. thick? 

Ans. 250560 bricks. 


Note. —It must be remembered that the side-walls take 
up 2 ft. from each end. Then we have two side-walls, 80 
ft. ( = 160 ft.), and two end-walls, 36 ft. ( = 72 ft.); so in 
the four walls we have one wall 232 ft. long, 20 ft. high 
and 2 ft. thick. 

5. Hon. Joseph E. Brown wishes to erect a 
building with bricks 8 in. long, 4 in. wide and 



38 


miller’s business arithmetic. 


3 in. thick, the building to be 210 ft. long, 
40 ft. wide and 30 ft. high. Leaving J for 
corners, doors and windows, how many bricks 
will be required, allowing the walls to be 2 ft. 
thick? Ans. 432000 bricks. 

To Find the Number of Bricks required to Pave 
a Yard or a Walk. 

Rule 19. Place the length and the width of 
the yard , in feet, with lJ^, on the right; on the 
left , the length and the width of the bricks , in 
inches. 

Prop. 1 . How many bricks 8 by 4 in. will 
be required to pave the floor of the court-house 
in Oglethorpe, Ga., the edifice being 80 ft. long 
and 40 ft. wide? Ans. 14400 bricks. 

OPERATION. 

i 0 

144x10x80 = 14400. 

Note.—A s the dimensions of the bricks are in inches 
and for surface measure, we use the 144 extra on the right 
to bring the surface to square inches: 144 sq. in. = 1 sq. ft. 

2. How many bricks 6x6 in. will be re- 



SHINGLES OR BOARDS TO COYER A ROOF. 39 


quired to pave a walk 1200 yd. (or 3600 ft.) 
in length and 16 ft. wide? 

Am. 230400 bricks. 

Note. —Always reduce the length and the width to feet. 

3. How many bricks 8 X 4 in. would it take 
to pave a space 47 miles long and 4 ft. wide ? 

Am. 4466880 bricks. 

To Find the Number of Shingles or Boards that 
will Cover a Roof. 

Rule 20. Place on the right the length of the 
roof, in feet, with double the length of rafters, 
in feet, also lJ/Jf.; on the left the width of the 
shingles or boards and the number of inches ex - 
posed to the weather. 

Prop. 1. How many shingles 4 in. wide,, 
with 6 in. exposed to the weather, will be re¬ 
quired to cover a house 30 ft. long, having 
rafters 12 ft, long? Am. 4320 shingles. 

OPERATION. 

i\ 30 

?! u 

I 144x30 = Ans. 4320. 

Note.— The 144 on the right reduces the square con¬ 
tents of the roof to square inches : 144 sq. in. = 1 sq. ft. 


40 miller's business arithmetic. 


2. How many shingles will it take to cover 
a house 60 ft. long, rafters to be 23 \ ft., shingles 
3^ in. wide, with 6 in. exposed to the weather ? 

Ans. 19337 + shingles. 

3. I wish enough boards to cover a barn 40 

ft. long, with rafters 18 ft., the boards to average 
6 in. wide and 8 in. exposed; how many will 
be required? Ans, 4320 boards. 

4. How many shingles 3J- in. wide and 5|- 
in. exposed to the weather will be required for 
a building 55 ft. long, the rafters 17^- ft. long? 

Ans. 14400 shingles. 

Note.—A fter adding the length of the rafters, in feet, 
if the sum be a mixed number, it must be reduced to an 
improper fraction, as all others. 


For Cutting Rafters. 

Rule 21. Place the width of the house and 3 
on the right; on the left , 5. 

Prop. 1. How long should a rafter be cut 
for a house 30 ft. wide? Ans. 18 ft. 

OPERATION. 

l 3x6 = 18 ft. 


CARPETING TO CARPET A FLOOR. 41 

Note. —While we take 3 ft. of the end-plate, we take 
5 ft. of rafter; then place 3 on the right and 5 on the 
left. We can give no perfect rule, as some desire one 
pitch and some another, though the one above is the most 
general. 

2. We wish rafters for a building 40 ft. wide; 
what should be the length ? Ans. 24 ft. 

3. What should be the length of rafters for 

a building 24 ft. wide? Ans. 14f ft. 

Note.— § of 1 ft. = 4f in., because § of 12 = 4f. 

4. J. L. Hand erects a mill-shed 80 ft. long 
and 30 ft. wide, and wishes a shingle roof; how 
many shingles 4 in. wide, with 6 in. exposed to 
the weather, would be required ? and what the 
cost, at $2^ per M. ? 

Ans. 17280 shingles; cost, $38.88. 

Note. —The rafters will be 18 ft. long by Rule 21; then, 
by Rule 20, the desired number of shingles can be found. 


To Find the Number of Yards of Carpeting 
needed to Carpet a Floor. 

Rule 22. Place the length and the width of 
the floor, in feet, with If, on the right; on the left, 
the width of the carpeting, in inches. 


42 miller’s BUSINESS ARITFIMETIC. 


Prop. 1. How many yards of carpeting 36 
in. wide will be required to carpet a floor 20 
ft. long and 18 ft. wide? Ans. 40 yd. 

OPERATION. 

n n 

| 4x10 = 40. 

Note. —It will be observed that 36 in. = - 3 ^ = 9 sq. ft. 
Thus, 40 in. wide = ~f- = 10 sq. ft., and so on to any width. 

2. How much carpeting 11 yd. (or 45 in.) 
wide will be required to carpet a hall 90 ft. 
long and 45 ft. wide? and what will be the 
cost, at $2J per yd. ? 

Ans. 360 yd.; cost, $810. 

3. How many yards of carpeting 45 in. wide 
will be required to carpet a floor 21 ft. 3 in. 
(= 211 ft.) long, 13 ft. 6 in. (= 131 ft.) wide? 

Ans. 25J- yd. 

4. What number of yards of carpeting } yd. 
wide will carpet a floor 30 ft. long, 20 ft. wide? 
and what will be the cost, at $lf ? 

^4ns. 88f yd.; cost, $155.55 +. 

5. How many yards of bagging 60 in. wide 
will carpet a court-room 40 X 60 ft. ? 

Ans. 160 yd. 


FOR COMPUTING INTEREST. 


43 


For Computing Interest. 

Rule 23. Place the principal, the number of 
months and the rate per cent, on the right; on 
the left, 12, or 3 and If days be expressed, 
place the principal, the number of days and the 
rate per cent, on the right; on the left, 360, or 
the factors J+, 9, 10. If the principal be dollars 
only, point off 2; if dollars and cents, point 

off 4~ 

Prop. 1 . What is the interest of $400 for 2 
mo., at 6 per cent, per annum? Ans. $4. 


OPERATION. 


n 


4.00 = 4.00. 

% 

0 


Note. —We reason thus: If 12, on the left, which rep¬ 
resents the time one year, will give the given rate per 
cent., on the right, what would the given principal and 
the given time, on the right, produce? In the above 
proposition we say, If 12 mo., on the left, will gain 6 
cts., what would $400 produce in 2 mo. ? 

When we work by days, we reason thus: If 360 days, 
on the left, will give the given rate per cent., on the 
right, what would the given principal and the given 
number of days, on the right, produce? 

2. What is the interest on $480 for 2 yr. 
and 6 mo., at 8 per cent. ? Ans. $96. 



44 miller’s business arithmetic. 


3. Give the interest on $4750 for 9 mo., at 

6 per cent. Airis. $213.75. 

4. What is the interest on $480 for 3 yr. 2 

mo., at lOf per cent. ? Ans. $163.40. 

5. Give the interest on $10,000 for 3 mo., at 

8 per cent. Ans. $200. 

6. What is the interest of $60 for 1 yr. 8 

mo., at 7 per cent. ? Ans. $7.00. 

7. At 7 per cent, for 2\ yr., what is the in¬ 
terest of $40 ? Ans. $7.00. 

8. What is the interest on $400 for 300 days, 

at 9 per cent. ? Ans. $30. 

OPERATION. 

$8$ | m 

m 300 x 10 = 30,00 = Ans. 

I 9 

Note. —If 360 days, on the left, will give 9 cents, what 
would $400 give in 300 days ? 

9. What is the interest on $180 for 400 days, 

at 6 per cent,? Ans. $12. 

10. Give the interest on $360 for 2 yr. 3 mo. 

20 da., at 8 per cent, Ans. $66.40. 

Note.— In 2 vr. and 3 mo. there are 27 mo. = 810 days, 
which, with 20 days, = 830 days. Then, if 360 days will 
give 8 cents, what would $360 give in 830 days ? When¬ 
ever days are expressed—years, months and days—the 
time must be reduced to days before placing on the line. 


TIME FROM ONE DATE TO ANOTHER. 45 


11. Give the interest on $24,000,000 for 1 
da., at 3 per cent, per annum. Ans. $2000. 

12. Supposing William H. Vanderbilt’s 
wealth to be $200,000,000, what is it worth 
for 1 da., at 6 per cent, per annum ? 

Ana. $33,333.33f 

13. What is the interest on $42 for 8 mo., at 

10 per cent. ? Ans. $2.80. 

Note. —When the amount is wanted, the principal and 
the interest are added together, and the sum equals the 
amount. 

14. What is the amount of $180, at 8 per 
cent., for 2 yr. 3 mo. and 20 da. ? 

Ana. $213.20. 

OPERATION. 

$33.20 Int. 

180.00 Prin. 

$213.20 Amt. 

To ascertain the Time that elapses from One 
Date to Another. 

Rule 24. Place the latest given date under 
the earliest—year under year , month under month , 
days under days , reckoning 12 months to the year 
and SO days to the month—and then work under 
the explanatory note. 



46 miller’s business arithmetic. 

Prop. 1. What time elapses from March 14, 
1858, to August 8, 1860? 

Ans. 2 yr. 4 mo. 24 da. 

OPERATION, 
yr. mo. da. 

1860 8 8 

1858 3_14 

2 4 24 

Note. —The latest given date is 1860, the 8th month 
^nd 8th day ; the earliest, 1858, 3d month, 14th day. The 
year 1858 is placed under 1860, the 3d month under the 
8th month, and 14 days under 8 days. We cannot take 
14 days from 8 days, so we borrow 1 month (30 days) and 
put it to the 8 days, which gives us 38 days. We then 
subtract 14 days from 38 days, which leaves us 24 days. 
It is proper now to pay back the 1 month borrowed, so 
we carry it to the 3d month, and say, “ 4 months from 8 
months leaves 4 months,” and set the 4 months under 
months. Then, 1858 from 1860 leaves the answer, 2 yr. 
4 mo. 24 da. 

2. A person was born on the 18th of De¬ 
cember, 1835, and died on the 12th of August, 
1875 ; what was his age at death? 

Ans. 39 yr. 7 mo. 24 da. 

OPERATION, 
yr. mo. da. 

1875 8 12 

1835 12 18 


39 


7 24 




INTEREST FROM ONE DATE TO ANOTHER. 47 

Note. —We cannot take 18 days from 12 days, but we 
borrow 1 month, or 30 days, and put to 12 days, which 
gives us 42 days. We then say, “18 days from 42 days 
leaves 24 days.” Now, as we borrowed 1 month from the 
8 months, it left the 8 months only 7 months. We now 
see that we cannot take 12 months from 7 months, so we 
borrow 1 year, or 12 months, and put to 7 months, which 
gives 19 months. We then say, “ 12 months from 19 
months leaves 7 months.” Now, as we have taken 1 year 
from 1875, 1874 is left. Then, 1835 years from 1874 years 
leaves 39 years. 

3. America, it is said, was discovered October 
11, 1492; how long to September 1, 1886? 

Ans. 393 yr. 10 mo. 20 da. 

Interest from One Date to Another. 

Prop. 1 . Give the interest on a note of $480 
from June 19, 1885, to July 1, 1886, at 7 per 
cent. Ans. $34.72. 

2. What is the interest on a note made on 
the 2d of January, 1885, for $900, and paid 
on the 30th of November, 1885, at 7 per cent. ? 

Ans. $57.40. 

3. What is the interest on a note of $6000 

for 6 days, at 6 per cent. ? Ans. $6.00. 

4. What is the amount of $600 for 6 days, 
at 6 per cent. ? 


48 miller’s business arithmetic. 


OPERATION. 

.60 Int. 

$600.0 0 Prin. 

$600.60 Amt. 

5. Give the interest on $360 for 6 days, at ; 

18 per cent. Ans. $1.08. 

6. What is the interest on $1200 for 18 days, 

at 2 per cent, per month (= 24 per cent, per 
annum) ? Ans. $14.40. 

Note.— f per cent, per mo. = 9 per cent, per annum. 

1 “ “ = 12 “ « 

li « « = 15 

H “ “ =18 “ « 

2 “ u =24 u « 

2i “ “ = 30 « « 

7. What interest is due A. J. Moye on a 

note of $600 which is overdue 45 days, at 2^- 
per cent, per month ? Ans. $22.50. 

8. Give the interest on $18 for 18 days, at ^ 

18 per cent, per annum. Ans. 16^ cts. 

Bank Discount. 

Bank discount is only simple interest paid in 
advance and reckoned for 3 days more than the 
given time. These 3 days are called “ days of 
grace.” i 





BANK DISCOUNT. 


49 




Rule 25. Place the principal, the number of 
days (with 3 days annexed ) and the rate per cent, 
on the right; on the left, 360, or 1+, 9, 10 

Prop. 1. What is the bank discount on 
$360, payable in 33 days, at 6 per cent.? 

Ans . $2.16. 


OPERATION. 


m 


m 

36 

6 

2.16 


Note. —The 33 days, with the 3 days of grace added, 
equal 36 days. Bank discount is worked, like interest, by 
days, except that 3 days are added for grace. 


2. What is the bank discount on $1000, pay¬ 
able in 90 days, at 18 per cent. ? 

Ans. $46.50. 

3. What is the present worth of a note of 

$360, due 77 days hence, discounted at bank at 
6 per cent. ? Ans. $355.20. 

Note. —When the present worth is wanted, the discount 
is subtracted from the principal, and the remainder is the 
present worth. 

4. What is the bank discount of a note of 
$36,000, due 97 days hence, discounted at 18f 

4 




50 miller’s business arithmetic. 


per cent.? and what is the present worth of 
the note? 

Am. The discount, $1875, taken from the 
principal, = $34,125, present worth. 

5. Give the bank discount on $5000, due in 

15 days, at 6 per cent. Ans. $15. 

6. What is the bank discount on a draft of 
$360, for 4 mo. 15 da., at 6 per cent. ? 

Ans. $8.28. 

Commercial Discount. 

Rule 26. Get the amount of 100 cents for the 
given time, at the given rate per cent., and place 
the sum or its factors on the left; on the right , 
the principal and 100, or 10, 10. 

Prop. 1. What is the present worth of a 
note for $1070, due 12 mo. hence, discounted 
at 7 per cent, per annum ? Am. $1000. 

OPERATION. 

I 1 

n\n 

| 7x1 = 7 cents. 

Note. We see that $1 at 7 per cent, is worth $1.07; 
this is added to 100, = 107, which goes on the left; the 
principal, $1070, with 100, on the right. We then reason 
thus: “If 107 cents, on the left, will give 100 cents, on 


TO FIND PERCENTAGE. 


51 


the right, what will the principal, $1070, on the right, 
give?” Thus: 

W I 100x10 = $1000. 

I xm 

2. What is the present worth of a note for 
$228, due 2 years hence, at 7 per cent. ? 

Ans. $200. 

Note.—H ere we see that in 2 years $1 at 7 per cent, is 
worth 14 cents. This added to 100=114, on the left, as a 
divisor. 

3. Capt. C. F. Hill, of Oglethorpe, Ga., gives 
me his note for $640.80, due 16 mo. hence; 
what is the present worth of said note, dis¬ 
counted at 6 per cent. ? Ans. $593.33|-. 

Note. —To get the discount, subtract the present worth 
from the principal. 

4. What is the discount of $3216, due 1 mo. 
6 da. hence, at 5 per cent. ? 

Ans. Present worth, $3200; discount, $16. 

To Find the Percentage on any Sum or Number 
at any Rate per cent. 

Rule 27. Place the principal {or sum) and 
the rate per cent, on the right; on the left, 100, 
or 10, 10. 


52 miller’s business arithmetic. 


Prop. 1. What is 8 per cent, of $200? 

Ans. $16. 

OPERATION. 

m | m 

I 8x2-16. 

Note. —If 100 cents, on the left, will give 8 cents, on 
the right, what will $200, on the right, give ? In all propo¬ 
sitions in percentage the 100 equals the number of cents 
in $1, and the 12 equals the number of months in a year. 

2. Messrs. B. K. Arthur & Son, of Shellman, 
Ga., bought 1000 bu. of oats, and sold 60 per 
cent, of the lot; how many bushels were sold ? 

Ans. 600 bu. 

3. What is 16§ per cent, of $18f ? 

Ans. $3.12J. 

4. What is 18f per cent, of $333^? 

Ans. $62.50. 

5. What is 33J per cent, of $33^? 

Ans. $111. 

6. Troup Amos has a note on B for $1000; 
B agrees to pay 33-^ per cent, of it in 12 mo., 
33^ per cent, of the remainder in 24 mo.; what 
did B pay each year, not allowing interest? 

Ans. 1st yr., $333,331; 2d yr., $222.22f; 
total, $555.55f 

7. What is 37^- per cent, of $750? 

Ans. $2811 


TO FIND RATE PER CENT. 53 

Having the Principal, the Time and the Amount 
Gained, to Find the Rate per cent. 

Rule 28. Place the principal and the time , 
in months, on the left; on the right , the amount 
gained, with 100 , 12. 

Prop. 1. I gave William G. Bateman, of 
Butler, Ga., $48 for the use of $600 for 1 yr.; 
what was the rate per cent. ? Ans. 8 per cent. 

OPERATION. 

0P0 = 8 per ct. 

n \ m 
v n 

Note.— We say, “If $600 in 12 mo., on the left, will 
gain $48, on the right, what will $100 gain in 12 mo.?” 

2. A paid B $48 for the use of $600 for 8 
mo.; what was the rate per cent. ? 

Ans. 12 per cent. 

3. Wesley Hill paid $8.82 for the use of $72 
for 1 yr. 9 mo.; what was the rate per cent. ? 

Ans. 7 p t cent. 

Note.—W hen figures are used for cents on one side of 
the line, ciphers must be used on the other. 

4. A agreed to pay B the sum of $5 for the 

use of $25 for 3 mo.; what was the rate per 
cent. ? Ans. 80 per cent. 


54 miller’s business arithmetic. 


5. I paid on $350, for 20 days, $3.50; what 
was the rate per cent.? Ans. 18 per cent. 

Note. —When days are expressed for time, aliquot part 
of a month must be used or taken. 10 da. = |mo.; 15 
da. = § mo.; 20 da. = § mo., etc. 

Having the Principal, the Rate per Cent, and the 
Amount Gained, to Find the Time. 

Rule 29. On the left place the principal and 
the rate per cent.; on the right , the amount gained , 
with 100 , 12. 

Prop. 1. How long must $140 be on interest 
at 6 per cent, to gain $42 ? Ans. 60 mo. 

OPERATION. 

m\ n a 

?\w 

20 I 12x5 = 60 mo. 

Note. —If 100 cents will gain in 12 months 6 cents, how 
long will it take $140 to gain $42? If the time is desired 
in years, place an extra 12 on the left. 

2. Prof. Howard W. Key, of Cuthbert, Ga., 
loaned me $500 and owes me $45; how long 
should I use the $500 at 9 per cent, to get. the 
worth of the $45? Ans. 12 mo. 




TO FIND THE PRINCIPAL. 


55 


3. If I advance B $35 as interest on $500 at 

10 per cent., how long am I to use it to get the 
worth of $35 ? Ans. 8f mo. 

4. C. A. J. Pope advanced $18.50 for the use 

of $185 at 12-J- per cent.; how long should he 
use the money ? Ans. 9f mo. 

5. I advance $180 for the use of $1800 at 

12 per cent.; how long can I use it for the 
$180? Ans. 10 mo. 

Having the Time, the Rate per cent, and the 
Amount Gained, to Find the Principal. 

Rule 30. Place on the right the amount 
gained and 100 , 12; on the left , the rate per 
cent, and the given time , in months. 

Prop. 1. What principal at 7 per cent, would 
be required to draw $700 in 12 mo. ? 

Ans. $10,000. 

OPERATION. 

j I M 

w , 100x100 = 10,000. 

I n 

Note.— If in 12 months 100 cents will gain t he given 
per cent., what principal will be required to draw the 
given amount in the given time? 

2. What principal at 10 per cent, would 
draw $1000 in 20 mo. ? Ans. $6000. 



56 miller’s business arithmetic. 


3. How much capital at 2 per cent, per annum 

would be required to draw $30,000 in 30 yr., 
or 360 mo. ? Ans. $50,000. 

4. A left his son an estate that at 7 per cent, 
paid him annually (every 12 mo.) $1200; what 
was the estate worth ? Ans. $17,142.85f-. 

5. What principal at 6 per cent, per annum 

would be required to draw $33,333.33^ in 12 
™o. ? Ans. $200,000,000. 

6. What principal at 9 per cent, will gain 

$84 in 1 yr. 9 mo. ? Ans. $560. 

Profit and Loss. 

Having the cost and the selling-price of an 
article, to find the gain or loss per cent. 

Rule 31. Place the cost on the left; on the 
right, the difference (between cost and selling-price) 
and 100. 

Prop. 1. I bought goods at 8 cts. per yd., 
and sold them at 10 cts. per yd.; what was the 
gain per cent. ? Ans. 25 per cent. 

OPERATION. 
t\W = 25. 

Note— If 8 cents, on the left, will gain 2 cents, on the 
right, what would 100 cents, on the right, gain? The 
above rule covers “Profit and Loss,” and is simple. 




GAIN PER CENT. 57 

2. We bought tobacco at 75 cts. per lb., aud 

sold the same at $1.00 per lb.; what was the 
gain per cent. ? Am. 33^ per cent. 

3. We bought tobacco at $1.00 per lb., and, 
being damaged', we sold it at 75 cts. per lb.; 
what was the loss per cent. ? Am. 25 per cent. 

4. Bought bacon at 10 cts. per lb., and sold 
it at 12^- cts.; what was the gain per cent. ? 

Am. 25 per cent. 

5. Mr. Isaac Early bought a quantity of 

sugar at 12^- cts. per lb.; he found it damaged, 
and sold it at 9-J- cts.; what was his loss per 
cent. ? Am. 24 per cent. 

6. Bought candy at 61 cts. per lb., and sold 
it at 18f cts.; what per cent, was gained ? 

Am. 200 per cent. 


Having the Cash- and the Credit-Price of an 
Article, to Find the Gain per cent. 

Rule 32. Place the cash-price and the given 
time , in months , on the left; on the right , the 
difference (between cash-price and credit-price), 

with ioo, m 

Prop. 1 . A sells his meat at 8 cts. per lb. 


58 miller’s business arithmetic. 


cash, but on 4 mo. time lie charges 10 cts. per 
lb.; what is the per cent, per annum ? 

Ans. 75 per cent. 

Note. —We say, “ If 8 cents in 4 months, on the left, 
will gain A 2 cents, what would 100 cents gain him in 12 
months?” Ans. 75 (which will ruin any poor farmer). 

2. Ed McDonald asks for a horse $90 in 
cash, but on 3 mo. time he asks $100; what is 
the rate per cent, per annum ? 

Ans. 44|- per cent. ? 

Note. —Rule 32 is a very important one to those who 
are so unfortunate as to be obliged to purchase on time. 
Not one planter in ten ever thinks what per cent, he is 
paying for the time given. 

3. Flour is selling for cash at $6 per barrel; 
on 3 mo. time, $8 per barrel is charged; what 
is the per cent, per annum ? 

Ans. 133J per cent. 

4. Prints are worth 5 cts. cash ; on 6 mo. 
time the price is 6J cts.; what is the per cent. ? 

Ans. 50 per cent. 

5. Tobacco is worth 50 cts. per lb. cash; on 
8 mo. time it is worth 62^- cts.; what is the rate 
per cent, per annum ? Ans. 37|- per cent. 



TO GAIN ANY GIVEN PER CENT. 


59 


Having the Cost of an Article, to Find how it 
should be Sold to Gain any given per cent. 

Rule 33. Add the desired per cent, to 100, 
and place the total on the right, also the cost; on 
the lefty 100. 

Prop. 1. Bought goods at 10 cts. per yard; 
what must we price them at to gain 20 per cent,? 

Ans. 12 cts. per yd. 

OPERATION. 

;p|W = 12. 

m I 10 

Note. —If 100 cents, on the left, will give 120 cents, on 
the right, wliat will the cost, 10 cents, give ? 

2. Bought goods at 20 cts. per yard; how 
should I price them to gain 40 per cent. ? 

Ans. 28 cts. 

3. We bought carpeting at 50 cts. per yard ; 

for how much per yard must it be sold to gain 
33J per cent. ? Ans. 66f cts. 

4. Bought candy at 18 J cts. per lb.; what 
must be the selling-price to gain 33^ per cent. ? 

Ans. 25 cts. 

5. Bought a horse for $90; what must be 
the selling-price to gain 33^- per cent. ? 

Ans. $120. 


60 miller’s business arithmetic. 


6. Bought shoes at $1,621 per pair; what 
must I price them at to gain 12|- per cent.? 

Am. $1.82i| cts. 

To Arrive at the Average Cost of a Nest of 
Tubs, Trays, Trunks, etc. 

Rule 34. Divide the cost of the nest hy the 
smn of all the vessels. 

Prop. 1. We bought a nest of 6 trunks for 
$5.04; what was the cost of each, beginning 
at the least ? 

OPERATION. 

1 1x24= 24 cts. 

2 2 x 24 = 48 

3 3x24= 72 

4 4x24= 96 

5 5x24 = 1.20 

J> 6x24 = 1,44 

Sura = 21)5.04(24 cts. $5.04 

' 42 
84 
84 

Note.—T he quotient, after dividing the cost by the sum, 
will equal the cost of the first; this, multiplied by the size 
in rotation, will give the cost of each in proportion to size. 

2. F. M. Alison bought a nest of 9 tubs for 
$4.50 ; what was the cost of each tub in pro¬ 
portion to the size? 



CAUSE AND EFFECT. 


61 


OPERATION. 


1 

1 x 10 = 10 cts. 

2 

2x10 = 20 

3 

3x10 = 30 

4 

4x10 = 40 

5 

5x10 = 50 

6 

6x10 = 60 

7 

7x10 = 70 

8 

8x10 = 80 

9 

9x10 = 90 

45)4.50 

$4.50 


3. J. W. Stanford bought a nest of 6 dishes 
at $4; what did each dish cost in proportion to 
size? 


OPERATION. 


1 

21)400(19* 

19*xl = 

19* 

2 

21 

19* x 2 = 

38* 

3 

190 

II 

CO 

X 

In 

05 
«—( 

57* 

4 

189 

19* x 4 = 

76* Fractions 

5 

* 

19* x 5 = 

95* =!• 

6 


19* x 6 = 

IIM 


21 $4.00 

Cause and Effect. 


Note. —The whole science of numbers may be embraced 
under the above principle—“ Cause and Effect ”—and 
under this is worked by proportion ; and where there is 
no proportion, there is no mathematics. Common sense 
teaches us that wherever there be a cause there must be 
an effect, and in my long experience as a teacher of math¬ 
ematics I have found that under this principle I can im- 



62 miller’s business arithmetic. 


part instruction with much more ease than under any 
other. Indeed, it is common-sense analysis, and is ap¬ 
preciated by all when understood. In my Comvion-Sense 
Arithmetic (brought out in 1882) the principle is, I think, 
fully explained, and by it may be easily understood. We 
now propose working a few propositions under the above 
heading, which we think will explain itself. 

Prop. 1 . If 10 bundles of fodder weigh 20 
lbs., what will 400 bundles weigh ? 

Am. 800 lbs. 


DIAGRAM FOR MAKING STATEMENT. 


1st cause. 

w 

2d effect. 


1st effect. 


2d cause. 
400x2 = 800. 


Note. —In this we see that 10 bundles of fodder is the 
1st cause, and 20 lbs. the 1st effect. Then, what would be 
the effect of 400 bundles under the 2d cause? 


2. If 7 hats cost § 
of 11 hats? 


, what would be the cost 
$44. 


Note. —Here 7 hats is the 1st cause, $28 the 1st effect. 
Then, what would be the effect of 11 hats? 


3. If 4 yd. of cloth cost 
5 yd. cost? 

OPERATION. 

n 

5x4 = 20, 


, what would 
Am. $20. 




CAUSE AND EFFECT. 


63 


4. If 5 yd. of cloth cost $20, what would 4 

yd. cost? Arts. $16. 

Note. —Simple causes produce simple effects, compound 
causes like effects, and for the proof reverse the operation, 
as in Propositions 3 and 4. 

5. If 7 horses in 4 da. consume 28 bu. of 

oats, how long will it take 5 horses to consume 
60 bu. ? Ans. 12 da. 


OPERATION. 


7 

ft 

i 


W 

0 

12 



Note. —In this we see that 7 horses and 4 da. are the 
1st cause, 28 bu. the 1st effect; 5 horses the 2d cause, and 
60 bu. the 2d effect. Causes are placed opposite causes, 
effects opposite effects, and there will always be a blank 
term, sometimes on the right, sometimes on the left. If 
on the left, the answer will be on the right; but if the 
blank term be on the right, the answer will be on the left. 

6. If 36 men, working 9 hr. Qach day for 4 
da., dig a ditch 240 ft. long, 8 ft. wide and 6 
ft. deep, what would be the depth of a ditch, 
60 ft. long and 6 ft. wide, dug by 9 men in 3 
da., working 6 hours per da. ? Ans. 4 ft. 



64 miller’s business arithmetic. 


OPERATION. 



m i 

3 

0 ^ = 4 ft. 


W 9 
9 3 
3 


Note— Here we see that 36 men, working 9 hr. a day 
for 4 da., gives a cause; a ditch 240 ft. long, 8 ft. wide 
and 6 ft. deep the effect. Then the 2d cause is 9 men in 
3 da., working 6 hours per day, to give the 2d effect, which 
is a ditch 60 ft. long and 6 ft. wide. The depth is a blank 
which is on the left; hence the answer is on the right. 


7. If 8 cats eat 8 rats in 8 min., how many 
cats will it take to eat 400 rats in 200 min. ? 

Ans. 16 cats. 

Proof.— If 16 cats in 200 min. eat 400 rats, how many 
cals in 8 min. will eat 8 rats? Ans. 8 cats. 

8. If 8 horses in 52 da. eat 416 bu. of oats, 
how many bushels will 26 mules eat in 30 da., 
each mule eating T 5 g as much as a horse ? 

Ans. 300 bu. 

9. Col. McLendon, of Thomasville, contracts 

to grade 24 miles of railroad in 8 mo.; for this 
he employs 150 men, who work 5 mo. and 
complete only 10 miles of the road; how many 
more men must be employed to finish the road 
in the time agreed on ? Ans. 200. 



CAUSE AND EFFECT. 


65 


OPERATION. 

n m i9 

5 

14 3 

_5 

70 

_5 

350 

Note. —It will be seen that 150 men in 5 mo. build only 
10 miles; he then had 3 mo. more to work on, and 14 
miles to grade. Then, if 150 men in 5 mo., as a cause, 
build 10 miles, the effect, how many men in 3 mo. (2d 
cause) will be required to build 14 miles (2d effect) in 
3 mo. (2d cause)? 

A ns. 350 men. But he has already 150, so 150 men 
from 350 men leave 200 more men. 

(The next two propositions I take from Prof. 
Caldwell’s noble work.) 

10. If in 6 da. 17^ hours long 16 men with 
13 mules can cut 312 yd. in length, 15^ yd. in 
width and 8§ ft. deep, how many mules will 
be required with 6 men in 3f days of 16j-^- 
hours long to perform another excavation, 50f 
yd. long, 10^ yd. wide and 15^ ft. deep? 

Ans. 12 mules. 

Note. —All mixed numbers must be reduced to improper 
fractions, placing the numerators where the mixed numbers 
occur, with its denominator on the opposite side. 

5 




66 miller’s business arithmetic. 


OPERATION. 


6 

312 

0 

171 

151 

171=3 n 

16 

8f 

£ Vfi 

13 


n 


— 

3 3 

50f 

6 

— 

10| 

3f 

m 

151 

16H 

3 
12 
3 H 
0 


m ? 

3 3 

W 

W 3 


0 

n 

m 

i 

0 

3 


Note. —All numbers are canceled out but 12, under the 
2d cause, which = Ans. 12 mules. 

11. If 112 men in 11 da. of 11 hours each 

dig a trench of 7° of hardness, 232^ ft. long, 
3| ft. wide and 2| ft. deep, in how many days 
of 5-^ hours long will 56 men dig a trench of 
5° of hardr css, 465 ft. long, 5f ft. wide and 3^- 
ft, deep? Ans. 144 days. 

12. If v compositors in 16 da. of 14 hours 
each can set 20 sheets of 24 pages each, 50 
lines in a page and 40 letters in a line, in how 
many days of 7 hours each will 10 compositors 
set a volume, to be printed in the same type, 
containing 40 sheets, 16 pages in a sheet, 60 
lines in a page and 50 letters in a line? 

Ans. 32 days. 

13. If 33 ft. of 8-in. belting, 4-ply, cost $9, 




PARTNERSHIP. 


67 


how many feet of 6-in. belting, 3-ply, can be 
bought for the same amount? Ans. 58f ft. 

Proof.— If 58f ft. of 6-in. belting, 3-ply, will cost $9, 
how many feet of 8-in. belting, 4-ply, will cost the same? 

A ns. 33 ft . 

12. If 10 men in 24 da. make 120 pairs of 
shoes, how many men in 15 da. will it take to 
make 600 pairs? Ans. 80 men.* 

Proof. —If 80 men in 15 da. make 600 pairs of shoes, 
how many men in 24 da. will it take to make 120 pairs? 

Ans. 10 men. 


Partnership. 

Partnership is an association formed between 
two or more individuals for business purpbses. 
The profits or the losses are divided among 
them, and each partner’s stock is employed for 
the same time as that of the others. Problems 
in Partnership are solved by the principle of 
cause and effect, under the following rule: 

Rule 35. Find the total amount invested , and 
place the aggregate under the first cause ; under 
the 1st effect place the gain or the loss ; under the 
2d cause A’s stock , then B’s, etc., making a state¬ 
ment for each. 


68 MILLEIl’s BUSINESS ARITFIMETIC. 


Prop. 1 . Sandlin, Fulton & Helms bought 
a lot of land for $1500, and sold the same at 
a gain of $300. Helms put in $400 ; Fulton, 
$500; Sandlin, $600. What was the share of 
each in the gain ? 

Ans. Helms, $80; Fulton, $100; Sandlin, 

$ 120 . 


OPERATION. 


Helms. 


Fulton. 


Sandlin. 


xm 

m 

xm 

m 

xm 


■■= $80. 

0 

000 = $100. 

•! 


m 

i2o. 


Note. —We find the total amount of the investment to 
be $1500. Then, if the sum $1500, as the 1st cause, will 
gain $300, the 1st effect, what will Helms’s $400 gain? 
Ans.> $80. Again, if $1500 will gain $300, what will 
Fulton’s $500 gain? Ans. $100. If $1500 will gain $300, 
what will Sandlin’s $600 gain? Ans. $120. 

$120+$100+ $80 =--$300. 


2. A and B engage in trade. A’s capital was 
$200; B’s, $300. They gained $1000; what 
was each one’s share of the gain? 

Ans. A’s, $400; B’s, $600. 

3. James Baisden, John Watts and Goode 
Jordan carried to market 24 bu. of peas, for 
which they received $24. Baisden put in ^ of 





PARTNERSHIP. 


69 


the peas; Watts, |; Jordan, J; to how much of 
the receipts was each entitled ? 

Ans. Baisden, fll^; Watts, $7^-; Jordan, 

$5 t 7 ^. 

Note. —In the solution of such propositions we must 
add together the fractions, which in the above form more 
than the unit (=|f), which gives a cause. Then, if yf, 
as the 1st cause, will produce $24, the first effect, what 
would \ produce ? ^ \ ? etc. 


OPERATION.’ 

Baisden. 

13 12x12 = 144 —13 = $11 

u_ 

2 1 


Watts. 

13 12 x 8 = 96 —13 = $7 t \ 

H 

2 i 


13 


Jordan. 

12x6 = 72-13 = $5& 


i i 


4. A gentleman divided his estate among his 
three children, John, William and Mary. To 
John he gave ^; to William, ^; to Mary, 





70 miller’s business arithmetic. 


His estate consisted of 17 elephants. What 
was the share of each ? 

Arts. John, 9; William, 6 ; Mary, 2, = 17. 

Note. —Partners share and share alike in their gains, 
also in losses; so in the above 1 + 1 + 1 fail to equal the 
unit, as it is equal only to If. Then, if H, as a cause, 
equal the whole number, 17, what will \ be? 9. Again, 
if 11 = 17, what will 1 equal ? 6. Again, if H = 17, what 
will 1 equal ? 2. 

5. A, B and C purchased goods to the amount 
of $150, and gained $60. A paid $40; B, $50; 
C, $60. What was the gain of each ? 

Ans. A, $16; B, $20; C, $24. 


Barter. 

Barter is the exchange of one commodity for 
another, and the operation is simplified by cause 
and effect. 

Rule 36. Under the 1st cause place the price 
of the desired quantity; under the 1st effect , the 
given quantity ; under the 2d cause , its price . 

Prop. 1 . How many pounds of raisins at 
10 cts. per pound must be given for 50 lbs. of 
candy at 18 cts. per pound? Ans. 90 lbs. 


AVERAGING THE PRICE. 


71 


OPERATION. 

I 18x5 = 90. 

\w 


Note. —If 10 cts., as the 1st cause, will give 18 cts., the 
1st effect, what will 50 lbs., under 2d cause, produce? 

2. What number of yards of jeans at 80 cts. 

per yard is equal to 60 yd. of cassimere at 80 
cts. per yard? Ans. 60 yd. 

3. How many yards of goods at 18 J cts. per 

yard can I get for 18f doz. eggs at 6^ cts. per 
dozen ? Ans. 56^ yd. 

4. What number of yards of goods at 80 cts. 

per yard is equal to 60 yd. of other goods at 
80 cts. per yard ? Ans. 60 yd. 

5. How many bushels of corn at 80 cts. per 

bushel is equal to 320 bu. of wheat at $1.60 per 
bushel ? Ans. 640 bu. 


Averaging the Price of Cotton or any other 
Commodity. 

Rule 37. Place the total number of pounds r 
bushels, etc., or the factors, on the left; on the 
right, the total cost, in dollars and cents. 


72 miller’s business arithmetic. 


Prop. 1 . Col. Cutts, of Americus, Ga., bought 
cotton as follows: 

5000 lbs. at 9 cts. per. lb. = $450.00 


2500 

“ 10 

a 

= 250.00 

3000 

“ 11 

a 

- 330.00 

500 

“ 12 

u 

= 60.00 

11000 



$1090.00 


OPERATION. 

11 m I 1090.00 

I w- 

What is the average cost ? Ans. cts. 

2. Bought cotton as follows: 

1050 lbs. at cts. 

2150 “ Si “ 

800 “ 10J « 

What is the average price per pound for the 
above bill ? Ans. 8^- cts. 

3. Bought the following bill of tobacco: 

150 lbs. at 16f cts. per lb 

100 “ 33 £ “ 

250 “ 621 « 

What was the average cost of the bill per pound ? 

Ans. 42^- cts. 




TO FIND THE CAPACITY OF A TANK. 73 


4. T. R. Bloom, of Macon, Ga., bought cotton 
as follows: 

3700 lbs. at 8 cts. per lb. 

6500 “ 9 “ 

50000 “ 9J “ 

1700 “ 11 “ 

2900 “ 1% “ 

What was the average cost per pound ? 

Am. 9^ 2 9 9 7 6 . 


To Find the Capacity of a Tank in Gallons. 

Rule 38. Place the length, the width and the 
depth, in feet, with 1728, on the right; on the left, 
231, or its factors, 3, 7, 11. 

Prop. 1. The C. R. R. tank at Graves Station 
is said to be 22 ft. long, 7 ft. wide and 6 ft. deep; 
how many gallons of water will it contain ? 

Am. 6912 gal. 


OPERATION. 


3 

1 

n 


w 

p 

1728x2x2 = 6912. 


Note. —The length, the width and the depth, in feet, on 
the right, with 1728, reduce the tank to cubic inches, and 
231, or 3, 7, 11, equals the number of gallons. 



74 miller’s business arithmetic. 


2. Harper Black’s tan-vats being 7 ft. long r 
5-J- ft. wide and 3^- ft. deep, how many gallons 
of water will each contain? Ans. 864 gals. 


To Find the Capacity of a Round Tank in 
Gallons. 

Rule 39. Place twice the inside diameter, m 
inches, with its depth, in inches, and 11, on the 
right; on the left, 231, 2, 7, or 2, 7, 3, 11, 7. 

Proi\ 1 . Dr. J. W. Stanford’s oil-tank is 22 
in. in diameter and 35 in. deep; how many gal¬ 
lons of oil will it contain? Ans. 57^f gab 


OPERATION. 


? 

7 

3 

1 

xx 


W Ilx5x22 = 1210-21 = ^4n$. 
22 
W 
XX 


Note. —The latter 3, 7,11 are factors of 231, the num¬ 
ber of cubic inches in 1 gallon. The above rule is worked 
on the same principle as Rule 48, etc. 


Having the Length, to Find how much Land to 
cut off to give any Desired Number of Acres. 

Rule 40. Place the given length of the land, 
in yards, on the left; on the right, the desired 
number of acres and 11, 11, 10, 



WIDTH OF AN ACRE IN YARDS. 75 


Prop. 1. We have a field 880 yd. long, and 
I rent to a tenant 8 A.; how far will I have to 
go to get the 8 acres ? 

OPERATION. 

m n 

ffl 11 A ns. 44 yd. 

w 

4x11-44 yd. 

2. How far would we go across a field 550 
yd. in length to get 11 A. ? Ans. 96f yd. 

3. How far would we have to go across a lot 
of land 990 yd. in length to get 45 A.? 

Ans. 220 yd. 

4. A lot of land 1100 yd. square contains 

250 A., but I sell my neighbor 75 A. on the 
south side of the lot; how wide a piece of land 
1100 yd. long will be required to produce the 
75 A. ? Ans. 330 yd. 

Having the Length in Yards, to Find the Width 
required for One Acre of Land in Yards. 

Pule 41. Place the length of the field, in 
yards, on the left; on the right, 11, 11, 10, If. 

Prop. 1. Suppose a field to be 440 yd. long; 
how wide a strip of land will be required for 
1 A.? Ans. 11 yd. 



76 


miller’s business arithmetic. 


OPERATION. 


m 

i 


11 

n 

w 

i 


Note.—A s 4840 sq. yd. = 1 A., we divide by the length, 
which equals the width required. 

2. W. J. Oliver has a field 550 yd. in length ; 
how wide a strip is required for 1 A. ? 

Ans. 8-| yd., or 26 ft. 8£ in. 

3. We have a tract of land just 1 mi. (or 
1760 yd.) in length; how wide a strip across 
this tract would be required for 1 A. ? 

Ans . 2-J yd. 

4. A owns a small patch of laud 80 yd. in 

length; what width would be required to equal 
1 A. across this patch ? Ans. 60^- yd. 

5. My neighbor John Mathews has a field 
220 yd. in length; how many rows 3 ft. apart 
would be required to equal 1 A. ? 

Ans. 22 yd., or 22 rows. 


An Easy and Quick Process for Multiplying any 
Number by the Aliquot Part of 100. 

Rule 42. If it is desired to multiply any 
number by 6\, annex two ciphers to the multipli¬ 
cand and divide by 16 or by its factors, because 
6 J is equal to of 100 . 




AN EASY AND QUICK PROCESS. 77 

Prop. 1 . Multiply 64 by 6£. Am. 400. 

To multiply any number by 12^: Annex 
two ciphers and divide by 8, because 12J = -§- 
of 100. 

To multiply by 16|: Annex two ciphers and 
divide by 6, because 16f = of 100. 

To multiply by 25 : Annex two ciphers and 
divide by 4, because 25 =£ of 100. 

To multiply by 33^-: Annex two ciphers and 
divide by 3, because 33^ = ^ of 100. 

To multiply by 37|-: Annex two ciphers and 
place the sum and 3 on the right and 8 on the 
left, because 37|- = f of 100. 

To multiply by 62^-: Annex two ciphers and 
place the sum on the right, with 5; on the 
left, 8. 

To multiply by 87-|: Annex two ciphers and 
place the sum on the right, with 7; on the 
left, 8. 

To multiply any number by 75 : Annex two 
ciphers and place the sum on the right, with 3; 
on the left, 4. 


78 miller’s business arithmetic. 

MENSURATION. 

The following geometrical rules are offered 
for the convenience of finding areas, etc., and 
will be found very nearly correct—that is, for all 
practical purposes. As for reasons, the student 
will have to search them out in the higher 
branches, without a knowledge of which he 
would fail to comprehend them were we to 
consume space in giving them. 

Having the Diameter of a Circle, to Find its Area. 

Rule 43. Place the diameter twice, with 11, 
on the right; on the left, 2 and 7. 

Prop. 1. What is the area of a circle 28 ft. 
in diameter? Ans. 616 sq. ft. 

OPERATION. 

?; 2j* 

* w 

I 11 x4xl4 = 616. 

Note.—= the unit; T 3 ¥ is taken up from the square 
for the 4 chords and the 4 arcs, lienee the square, 

multiplied by 4 $, = its area. If you work in inches, the 
result will be square inches; if in feet, square feet; if in 
yards, square yards; if in miles, square miles. 

2. We have a circle 63 yd. in diameter; what 
is its area in square yards ? Ans. 3118J sq. yd. 


MENSURATION. 


79 


3. We have a circular garden 140 yd. in 
diameter; what is its area in square yards and 
square acres ? Ans. 15400 sq. yd., or A. 

Note. —When the area in acres is asked for, divide the 
result by 4840, or place 11, 11, 10, 4 extra on left—that 
is, for yards. If rods, put 4, 10, 4 on the left, extra; if in 
chains, put 10 extra on the left, etc. 

4. The city of Dawson is incorporated 1 m. 
each way from the court-house; how many acres 
are included in the corporate limits? 

Note. —We find that the corporate limits contain 3f 
sq. m., and as 1 m. = 640 A. the 3| equal the answer. In 
miles put 640 extra, on the right, to get the result in 
acres. 


Having the Circumference of a Circle, to Find 
its Area. 

Rule 44. Place the circumference twice and 
7 on the right; on the left , 8, 11. 

Prop. 1. A lake is 880 yd. in circumference; 
what is its area in square yards ? 

Ans. 61600 sq. yd. 

2. Lake Defeniak, in Florida, is 1 m., or 
1760 yd., in circumference; what is its area in 
acres ? Ans. 50ja A.- 


80 miller’s business arithmetic. 


OPERATION. 


% xx 

11 


xv 

XX 

X 


xm xw 
xm 

7x80 = 560-ll = ^ns. 


m 

XX 

X 


Note. —As the circumference is considered in yards, we 
place on the left 11, 11, 10, 4, besides 8, 11, which gives 
the result in acres. Had we worked in miles, we would 
have followed the rule for miles and put 640 extra for 
acres on the right. 


I 


SECOND operation. 

0 1 
11 1 
7 

fyXty 80 x 7 = 560 -r- 11 = the same result, 50H A. 


3. How many square feet in a circle whose 
outside measurement is 88 ft. ? 

Ans. 616 sq. ft. 

4. George Johnson has a meadow whose 

fencing measures 110 yd.; how many acres 
does it contain ? Ans. A. 


To Find the Area of a Globe or a Ball. 

Rule 45. On the right place the diameter 
twice and 22; on the left, 7. 

Prop. 1. What is the area of a globe 14 in. 
in diameter? Ans. 616 sq. in. 




MENSURATION. 


81 


OPERATION. 

■in* 

I 14 

22x2x14 = 616. 

Note. —If the diameter be in inches, the result will be 
square inches; if in feet, square feet. 

2. How many square feet on the surface of 
a globe or ball 3^- ft. in diameter? 

Ans. 38|- sq. ft. 

3. What is the area of a globe 10J ft. in 

diameter? Ans. 346| sq. ft. 

To Find the Solidity of a Globe or a Ball. 

Rule 46. Place the diameter thrice , with 11 9 
on the right; on the left, 3 9 7. 

Prop. 1. How many solid inches in a ball 
21 in. in diameter? Ans. 4851 sol. in. 

OPERATION. 

3 31 

1 21x21x11 = 4851. 

21 

II 

Note. —In solids, length, breadth and depth are con¬ 
sidered ; hence the cube, etc. Inches produce inches; feet 
produce feet; yards produce yards. 

6 



82 MILLER^ BUSINESS ARITHMETIC. 

2. We have a base-ball 3£ in. in diameter; 
how many solid inches in the ball ? 

Ans. 22^ sol. in. 

To Find the Area of the Largest Square that can 
be Inscribed within a Circle. 

Rule 47. Place the diameter twice on the 
right; on the left , 2. 

Prop. 1. What is the area of a square that 
can be inscribed in a circle 14 in. in diameter? 

Ans. 98 sq. in. 

OPERATION. 

| 14x7 = 98. 

Note.— Multiplying the square of any diameter by £ 
will produce its area. Then, to get one of the sides, the 
square root of its area must be extracted. For the mode 
of doing this reference is made to any writer on arithmetic 
who treats of square root. We cannot simplify nor ab¬ 
breviate either cube root or square root from the many 
authors who have preceded us; hence both are omitted 
in this manual. 

2. What is the area of a square that can be 
inscribed in a circle 60 ft. in diameter? 

Ans. 1800 sq. ft. 

3. Give the area of a circle 30 in. in diameter? 

Ans. 450 sq. in. 



MENSURATION. 


83 


To Find the Solidity of a Cylinder. 

Rule 48. Place its diameter twice , and also 
its lengthy with 11 } on the right; on the lefty 2, 7. 

Prop. 1. Required the solidity of a cylinder 
4 ft. in diameter and 10 ft. long. 

Ans. 125f sol. ft.. 

OPERATION. 

? i 
7 4 

10 

11 x2x 10 x 4-880-5-7 = 125f. 

Note. —If the diameter be in inches, the length must 
be taken in inches, and the result will be cubic inches; if 
feet, cubic feet. 

2. What are the solid contents, in feet, of a 
cylinder 2 ft. in diameter and 10 ft. long? 

Ans. 3If sol. ft. 

3. How many solid feet in a screw-pin 10^ 
in. in diameter and 16 ft. long? 

Ans. cu. ft. 

4. At 1^- cts. per solid inch, what would be 

the cost of a screw-pin 3^- in. in diameter and 
10 ft. long? Ans. $17.32f 

Note. —We see that the above pin gives 1155 cu. in.; 
this, at 11 cts. per cubic inch, gives the answer. 



84 miller’s business arithmetic. 


OPERATION. 


2 

? 

% 

71 

% 


7 3| = £ 

£20 10 ft. long = 120 in. 
11 l^cts. = fct. 

3 


15x3x11x7 = 3465 = 2 = the cost. 


6. How many cubic ft. in a round shaft 2 ^ 
ft. in diameter and 30 ft. long ? 

Ans. 147^g cu. ft. 


Having the Diameter of two Circles, one formed 
within the other, to Find the Area of the Ring. 

Rule 49. Place the sum of the diameters , with 
their difference and 11 , on the right; on the left, 
0, 7. 


Prop. 1. We have two circles, one 8 ft. in 
diameter, and the other 6 ft. in diameter; how 
many square feet between the two circles? 

Ans. 22 sq. ft. 



OPERATION. 


% 

71 


U 
2 . 

11 

22 sq. ft. 


8 + 6 = 14 sum of diameters. 
8 — 6 = 2 difference in “ 





MENSURATION. 


85 


Note.—T he above rule can be applied to finding the 
number of cubic feet of stone to wall up a round well by 
placing the depth of the well in feet extra on the right. 

2. What is the area of a ring formed by two 
circles 10 and 20 yd. in diameter? 

Ans. 235-f- sq. yd. 

3. Col. W. H. Willis, of Oglethorpe, Ga., 
has a round well dug, 80 ft. deep, 7 ft. in 
diameter, and walled up with stone so as to 
leave the diameter 3^ ft.; how many solid feet 
of stone were required? Ans. 2310 sol. ft. 

4. The C. R. R. digs a round well 80 ft. deep 
and 6 ft. in diameter, and walls it with stone 
so as to leave the opening 4 ft. in diameter; 
what number of cubic feet of stone is required ? 

Ans. 1257^- cu. ft. 


To Find where a Pole should be Broken to Strike 
the Ground at any Given Distance and Hang 
on the Stump. 

Rule 50. Square the height of the pole , to 
which add the square of the base , and place the 
sum on the right; on the left ,, the height of the 
pole and 2. 


86 miller’s business arithmetic. 

Prop. 1. On a level plain stands a pole 100 
ft. high; how much must break off to strike 
80 ft. from the stump and hang? 

Ans. 82 ft. 


OPERATION. 



m 

% 


mn 

m 

82 


100x100 = 10000 
80 x 80= 6400 

16400 = 

the sum on right. Height, 100, 
and 2, on left, =the divisor. 

Note. —If the question is, 
How high above the ground 
must the pole break off? sub¬ 
tract the amount broken off 
from the height of the pole. Thus, 100 — 82 = 18 ft. Or 
if you subtract the square of the base from the square of 
the height and divide the remainder by twice the height, 
this will also give the height of the stump. 


2. On a level plain stands a pole 80 ft. high ; 

at what height above the ground fnust it be 
broken to strike the ground 40 ft. from the 
stump and hang? Ans . 30 ft. 

3. On a level plain stands a pole 33 ft. 8 in. 
(= 332 . ft.) j how much must break off to strike 






MENSURATION. 


87 


the ground 20 ft. 4 in. (= 20-J ft.) from where 
it stands and hang to the stump ? 

Ans. 22«| ft. 

To Measure the Distance to any Given Object. 

Rule 51. Draw a right-angled triangle, A, E r 
D, B, SO ft. Stand at the point E and take C 
for the given object, and where it crosses the line 
BD (at F) stick a peg ; then c 
measure the distance, in feet, 
from F to D, and place it, 
in feet, on the left, and SO, SO 
on the right. This gives the 
distance from A to C in feet. 

Prop. 1. In the accom¬ 
panying diagram we find the 
distance from F to D to be 
2 ft.; what is the distance B 
from A to C ? Ans. 450 ft. 

OPERATION. 

30x15 = 450 ft. 

Note.— If the right angle be in 
yards, the result will be yards; if the angle be rods, the 
result will be rods. This angle must be measured accu¬ 
rately, and the distance from F to D must be accurate, to 








88 miller’s business arithmetic. 

obtain an accurate result. If F to D be taken in inches, 
AEDB must be brought to inches, and then the result 
from A to C will be inches. To get the distance from A 
to C, subtract AB from AC. 

2. What would be the distance from A to C 

if FJD be 3 ft.? Am. 300 ft. 

3. What would be the distance from A to C 

if FD be 2J ft. ? Am. 360 ft. 


PLANTS ON AN ACRE OF LAND 


89 


TABLE SHOWING THE NUMBER OP PLANTS ON AN ACRE OF LAND 
AT ANY GIVEN DISTANCE APART. 



No. Plants. 


No. Plants. 

9 ft. 0 in. by 9 ft. Oin. 

537 

5 ft 

6 in. by 4 ft. Oin. 

1980 

y U 

8 

0 

605 

5 

6 

3 

9 

2112 

9 0 

7 

0 

691 

5 

6 

3 

6 

2262 

. 9 0 

6 

0 

806 

1 5 

6 

3 

3 

2436 

9 0 

5 

0 

968 

5 

6 

3 

0 

2640 

8 0 

8 

0 

680 

5 

6 

2 

9 

2880 

8 0 

7 

0 

777 

5 

6 

2 

6 

3168 

8 0 

6 

0 

905 

5 

6 

2 

3 

3520 

8 0 

5 

0 

1089 

5 

6 

2 

0 

3960 

8 0 

4 

0 

1361 

5 

6 

1 

9 

4525 

8 0 

3 

0 

1815 

5 

6 

1 

6 

5280 

7 0 

7 

0 

888 

5 

6 

1 

3 

6336 

7 0 

6 

6 

957 

5 

6 

1 

0 

7920 

7 0 

6 

0 

1037 

5 

0 

5 

0 

1742 

7 0 

5 

6 

1131 

5 

0 

4 

9 

1834 

7 0 

5 

0 

1244 

5 

0 

4 

6 

1936 

7 0 

4 

6 

1382 

5 

0 

4 

3 

2049 

7 0 

4 

0 

1555 

5 

0 

4 

0 

2178 

7 0 

3 

6 

1777 

5 

0 

3 

9 

2323 

7 0 

8 

0 

2074 

5 

0 

3 

6 

2489 

7 0 

2 

6 

2489 

5 

0 

3 

3 

2680 

7 0 

2 

0 

3111 

5 

0 

3 

0 

2904 

7 0 

1 

6 

4148 

5 

0 

2 

9 

3168 

7 0 

1 

0 

6222 

5 

0 

2 

6 

3484 

6 6 

6 

6 

1031 

5 

0 

2 

3 

3872 

6 6 

6 

0 

1116 

5 

0 

2 

0 

4356 

6 6 

5 

6 

1218 

5 

0 

1 

9 

4978 

6 6 

5 

0 

1340 

5 

0 

1 

6 

5808 

6 6 

4 

6 

1489 

5 

0 

1 

3 

6969 

6 6 

4 

0 

1675 

5 

0 

1 

0 

8712 

6 6 

3 

6 

1914 

4 

9 

4 

9 

1930 

6 6 

3 

0 

2233 

4 

9 

4 

6 

2037 

6 6 

2 

6 

2680 

4 

9 

4 

3 

2157 

6 6 

2 

0 

3350 

4 

9 

4 

0 

2292 

6 6 

1 

6 

4467 

4 

9 

3 

9 

2445 

6 6 

1 

0 

6701 

4 

9 

3 

6 

2620 

6 0 

6 

0 

1210 

4 

9 

3 

3 

2821 

6 0 

5 

9 

1262 

4 

9 

3 

0 

3056 

6 0 

5 

6 

1320 

4 

9 

2 

9 

3334 

6 0 

5 

0 

1452 

4 

9 

2 

6 

3668 

6 0 

4 

6 

1613 

4 

9 

2 

3 

4075 

6 0 

4 

0 

1815 

4 

9 

2 

0 

4585 

6 0 

3 

6 

2074 

4 

9 

1 

9 

5248 

6 0 

3 

0 

2420 

4 

9 

1 

6 

6113 

6 0 

2 

G 

2904 

4 

9 

1 

3 

7336 

6 0 

2 

0 

3630 

4 

9 

1 

0 

9170 

6 0 

1 

6 

4840 

4 

6 

4 

6 

2151 

6 0 

1 

0 

7260 

4 

6 

4 

3 

2277 

5 6 

5 

6 

1417 

4 

6 

4 

0 

2420 

5 6 

5 

3 

1508 

4 

6 

3 

9 

2581 

5 6 

5 

0 

1584 

4 

6 

3 

6 

2765 

5 6 

4 

9 

1667 

4 

6 

3 

3 

2978 

5 6 

4 

6 

1760 

4 

6 

3 

0 1 

3226 























90 miller’s business arithmetic. 


TABLE SHOWING NUMBER OF PLANTS, Etc. ( Continued ). 



No. Plants. 


No. Plants. 

4 ft. 6 in. by 2 ft. 9 in. 

3520 

3 ft. 6 in. by 1 ft. 0 in. 

12145 

4 6 

2 

6 

3872 

3 

3 

3 

3 

4124 

4 6 

2 

0 

4840 

3 

3 

3 

0 

4818 

4 6 

1 

9 

5531 

3 

3 

2 

9 

4873 

4 6 

1 

6 

6453 

3 

3 

2 

6 

5361 

4 6 

1 

3 

7774 

3 

3 

2 

3 

5956 

4 6 

1 

0 

9680 

3 

3 

2 

0 

6701 

4 3 

4 

3 

2411 

3 

3 

1 

9 

7658 

4 3 

4 

0 

2562 

3 

3 

1 

6 

8935 

4 3 

3 

9 

2733 

3 

3 

1 

3 

10722 

4 3 

3 

6 

2914 

3 

3 

1 

0 

13403 

4 3 

3 

3 

3153 

3 

0 

3 

0 

4840 

4 3 

3 

0 

3416 

3 

0 

2 

9 

5289 

4 3 

2 

9 

3727 

3 

0 

2 

6 

5808 

4 3 

2 

6 

4099 

3 

0 

2 

3 

6453 

4 3 

2 

3 

4555 

3 

0 

2 

0 

7260 

4 3 

2 

0 

5124 

3 

0 

1 

9 

8297 

4 3 

1 

9 

5856 

3 

0 

1 

6 

9680 

4 3 

1 

6 

6832 

3 

0 

1 

3 

11616 

4 3 

1 

3 

8199 

3 

0 

1 

0 

14520 

4 3 

1 

0 

10249 

2 

9 

2 

9 

5760 

4 0 

4 

0 

2722 

2 

9 

2 

6 

6336 

4 0 

3 

9 

2904 

2 

9 

2 

3 

7040 

4 0 

3 

6 

3111 

2 

9 

2 

0 

7920 

4 0 

3 

3 

3350 

2 

9 

1 

9 

9051 

4 0 

3 

0 

3630 

2 

9 

1 

6 

10560 

4 0 

2 

9 

3960 

2 

9 

1 

3 

12670 

4 0 

2 

6 

4356 

2 

9 

1 

0 

15840 

4 0 

2 

0 

5445 

2 

6 

2 

6 

6969 

4 0 

1 

9 

6222 

2 

6 

2 

3 

7740 

4 0 

1 

6 

7260 

2 

6 

2 

0 

8712 

4 0 

1 

3 

8712 

2 

6 

1 

9 

9956 

4 0 

1 

0 

10890 

2 

6 

1 

6 

11616 

3 9 

3 

9 

3097 

2 

6 

1 

3 

13939 

3 9 

3 

6 

3318 

2 

6 

1 

0 

17424 

3 9 

3 

3 

3574 

2 

3 

2 

3 

8604 

3 9 

3 

0 

3872 

2 

3 

2 

0 

9680 

3 9 

2 

9 

4224 

2 

3 

1 

9 

11062 

3 9 

2 

6 

4646 

2 

3 

1 

6 

12106 

3 9 

2 

3 

5162 

2 

3 

1 

3 

15488 

3 9 

2 

0 

5808 

2 

3 

1 

0 

19360 

3 9 

1 

9 

6637 

2 

0 

2 

0 

10890 

3 9 

'1 

6 

7744 

2 

0 

1 

9 

12445 

3 9 

1 

3 

9272 

2 

0 

1 

6 

14520 

3 9 

1 

0 

11616 

2 

0 

1 

3 

17424 

3 6 

3 

6 

3555 

2 

0 

1 

0 

21780 

3 6 

3 

3 

3829 

1 

9 

1 

9 

14232 

3 6 

3 

0 

4148 

1 

9 

1 

6 

16594 

3 6 

2 

9 

4525 

1 

9 

1 

3 

19913 

3 6 

2 

6 

4978 

1 

9 

1 

0 

24454 

3 6 

2 

3 

5531 

1 

6 

1 

6 

19360 

3 6 

2 

0 

6222 

1 

6 

1 

3 

23232 

3 6 

1 

9 

7111 

1 

6 

1 

0 

29040 

3 6 

1 

6 

8297 

1 

3 

1 

3 

27878 

3 6 

1 

3 

9556 

1 

0 

1 

0 

43560 









































* 








































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